E2 VoC Analyses Part 3: RL Analyses

AIC average by age group

Examine age-related change in parameter estimates from models

Run regressions between model parameters and age

## 
## Call:
## lm(formula = LL ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -154.578  -58.515    9.859   54.104  200.722 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -215.354     19.598  -10.99   <2e-16 ***
## age            1.591      1.170    1.36    0.176    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 72.28 on 148 degrees of freedom
## Multiple R-squared:  0.01235,    Adjusted R-squared:  0.005673 
## F-statistic:  1.85 on 1 and 148 DF,  p-value: 0.1758

## 
## Call:
## lm(formula = alphaPosChoice ~ age, data = model_params)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.3090 -0.1999 -0.0917  0.1265  0.6863 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.320756   0.074344   4.314 2.91e-05 ***
## age         -0.001369   0.004438  -0.308    0.758    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2742 on 148 degrees of freedom
## Multiple R-squared:  0.0006424,  Adjusted R-squared:  -0.00611 
## F-statistic: 0.09514 on 1 and 148 DF,  p-value: 0.7582

## 
## Call:
## lm(formula = alphaNegChoice ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.20679 -0.12581 -0.06585  0.00345  0.81559 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.304868   0.061239   4.978 1.76e-06 ***
## age         -0.011916   0.003656  -3.260  0.00138 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2259 on 148 degrees of freedom
## Multiple R-squared:  0.06698,    Adjusted R-squared:  0.06068 
## F-statistic: 10.62 on 1 and 148 DF,  p-value: 0.001385

## 
## Call:
## lm(formula = alphaPosComp ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.23170 -0.17898 -0.12042  0.05715  0.87437 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.300954   0.073408   4.100  6.8e-05 ***
## age         -0.007638   0.004382  -1.743   0.0834 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2707 on 148 degrees of freedom
## Multiple R-squared:  0.02012,    Adjusted R-squared:  0.01349 
## F-statistic: 3.038 on 1 and 148 DF,  p-value: 0.08341

## 
## Call:
## lm(formula = alphaNegComp ~ age, data = model_params)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.2432 -0.2170 -0.1918  0.2049  0.7903 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  0.270545   0.088504   3.057  0.00265 **
## age         -0.003096   0.005284  -0.586  0.55877   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3264 on 148 degrees of freedom
## Multiple R-squared:  0.002315,   Adjusted R-squared:  -0.004426 
## F-statistic: 0.3434 on 1 and 148 DF,  p-value: 0.5588

## 
## Call:
## lm(formula = betaAgency ~ age, data = model_params)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -9.158 -4.012 -1.747  2.933 20.338 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  3.53843    1.51875   2.330  0.02117 * 
## age          0.24547    0.09067   2.707  0.00758 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.601 on 148 degrees of freedom
## Multiple R-squared:  0.04719,    Adjusted R-squared:  0.04075 
## F-statistic:  7.33 on 1 and 148 DF,  p-value: 0.007579

## 
## Call:
## lm(formula = betaMachine ~ age, data = model_params)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -7.833 -3.151 -1.048  1.911 22.856 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.9326     1.3383   3.686 0.000319 ***
## age           0.1402     0.0799   1.755 0.081273 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.936 on 148 degrees of freedom
## Multiple R-squared:  0.02039,    Adjusted R-squared:  0.01378 
## F-statistic: 3.081 on 1 and 148 DF,  p-value: 0.08127

## 
## Call:
## lm(formula = agencyBonus ~ age, data = model_params)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.72271 -0.30021 -0.19730  0.08013  2.37437 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.72607    0.18718   3.879 0.000157 ***
## age         -0.01312    0.01117  -1.174 0.242346    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6904 on 148 degrees of freedom
## Multiple R-squared:  0.009224,   Adjusted R-squared:  0.00253 
## F-statistic: 1.378 on 1 and 148 DF,  p-value: 0.2423

Plot relations between model parameters and age

Parameter summary statistics

Beta model

## Mixed Model Anova Table (Type 3 tests, S-method)
## 
## Model: estimate ~ ageZ * betaType + (1 | subject_id)
## Data: betas
##          Effect        df      F p.value
## 1          ageZ 1, 148.00 6.75 *    .010
## 2      betaType 1, 148.00   0.45    .503
## 3 ageZ:betaType 1, 148.00   1.55    .216
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: estimate ~ ageZ * betaType + (1 | subject_id)
##    Data: data
## 
## REML criterion at convergence: 1803.2
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.9989 -0.5011 -0.1486  0.4083  3.2037 
## 
## Random effects:
##  Groups     Name        Variance Std.Dev.
##  subject_id (Intercept) 14.21    3.770   
##  Residual               13.66    3.696   
## Number of obs: 300, groups:  subject_id, 150
## 
## Fixed effects:
##                Estimate Std. Error       df t value Pr(>|t|)    
## (Intercept)      7.3161     0.3745 148.0000  19.534   <2e-16 ***
## ageZ             0.9745     0.3752 148.0000   2.597   0.0103 *  
## betaType1        0.1433     0.2134 148.0000   0.672   0.5028    
## ageZ:betaType1   0.2658     0.2137 148.0000   1.244   0.2155    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) ageZ  btTyp1
## ageZ        0.000              
## betaType1   0.000  0.000       
## ageZ:btTyp1 0.000  0.000 0.000

Predictor	Estimates	SE	Statistic	p
intercept	7.32	0.37	19.53	<0.001
age	0.97	0.38	2.60	0.010
decision stage	0.14	0.21	0.67	0.502
age x decision stage	0.27	0.21	1.24	0.215

Learning rate model

## Mixed Model Anova Table (Type 3 tests, S-method)
## 
## Model: estimate ~ age_z * valence * agency + (1 | subject_id)
## Data: learning_rates
##                 Effect        df         F p.value
## 1                age_z 1, 148.00    5.38 *    .022
## 2              valence 1, 444.00 11.17 ***   <.001
## 3               agency 1, 444.00      0.10    .753
## 4        age_z:valence 1, 444.00      0.51    .477
## 5         age_z:agency 1, 444.00      0.09    .762
## 6       valence:agency 1, 444.00 28.34 ***   <.001
## 7 age_z:valence:agency 1, 444.00    3.20 +    .074
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: estimate ~ age_z * valence * agency + (1 | subject_id)
##    Data: data
## 
## REML criterion at convergence: 199.6
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.5327 -0.6127 -0.3271  0.2431  3.2261 
## 
## Random effects:
##  Groups     Name        Variance Std.Dev.
##  subject_id (Intercept) 0.008624 0.09287 
##  Residual               0.067885 0.26055 
## Number of obs: 600, groups:  subject_id, 150
## 
## Fixed effects:
##                          Estimate Std. Error         df t value Pr(>|t|)    
## (Intercept)              0.203361   0.013063 148.000000  15.568  < 2e-16 ***
## age_z                   -0.030316   0.013074 148.000000  -2.319   0.0218 *  
## valence1                -0.035555   0.010637 444.000000  -3.343   0.0009 ***
## agency1                  0.003345   0.010637 444.000000   0.314   0.7533    
## age_z:valence1          -0.007579   0.010646 444.000000  -0.712   0.4769    
## age_z:agency1           -0.003219   0.010646 444.000000  -0.302   0.7625    
## valence1:agency1        -0.056629   0.010637 444.000000  -5.324 1.62e-07 ***
## age_z:valence1:agency1  -0.019045   0.010646 444.000000  -1.789   0.0743 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) age_z valnc1 agncy1 ag_z:v1 ag_z:g1 vln1:1
## age_z       0.000                                            
## valence1    0.000  0.000                                     
## agency1     0.000  0.000 0.000                               
## age_z:vlnc1 0.000  0.000 0.000  0.000                        
## age_z:gncy1 0.000  0.000 0.000  0.000  0.000                 
## vlnc1:gncy1 0.000  0.000 0.000  0.000  0.000   0.000         
## ag_z:vln1:1 0.000  0.000 0.000  0.000  0.000   0.000   0.000

## 
##  Paired t-test
## 
## data:  model_params$alphaPosChoice and model_params$alphaNegChoice
## t = 6.9666, df = 149, p-value = 9.73e-11
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  0.1320728 0.2366607
## sample estimates:
## mean difference 
##       0.1843667

## 
##  Paired t-test
## 
## data:  model_params$alphaPosComp and model_params$alphaNegComp
## t = -1.1108, df = 149, p-value = 0.2685
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  -0.11712714  0.03283236
## sample estimates:
## mean difference 
##     -0.04214739

Predictor	Estimates	SE	Statistic	p
intercept	0.20	0.01	15.57	<0.001
age	-0.03	0.01	-2.32	0.021
valence	-0.04	0.01	-3.34	0.001
agency	0.00	0.01	0.31	0.753
age x valence	-0.01	0.01	-0.71	0.477
age x agency	-0.00	0.01	-0.30	0.762
valence x agency	-0.06	0.01	-5.32	<0.001
age x valence x agency	-0.02	0.01	-1.79	0.074

Learning rate plot

Relation between parameter estimates and ‘model-free’ regressions

## 
## Call:
## lm(formula = `(Intercept)` ~ agencyBonus, data = voc_REs_RL)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.9134 -0.6544  0.0318  0.7376  5.8069 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -1.2960     0.1436  -9.025 8.77e-16 ***
## agencyBonus   2.4096     0.1668  14.449  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.407 on 148 degrees of freedom
## Multiple R-squared:  0.5852, Adjusted R-squared:  0.5824 
## F-statistic: 208.8 on 1 and 148 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = voc_z ~ betaAgency, data = voc_REs_RL)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.13455 -0.22221 -0.03153  0.14601  1.68526 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.779560   0.049299  -15.81   <2e-16 ***
## betaAgency   0.102000   0.005251   19.42   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3666 on 148 degrees of freedom
## Multiple R-squared:  0.7182, Adjusted R-squared:  0.7163 
## F-statistic: 377.3 on 1 and 148 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = voc_z ~ betaAgency + age, data = voc_REs_RL)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.16014 -0.23331 -0.02059  0.18482  1.67324 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.967014   0.099995  -9.671   <2e-16 ***
## betaAgency   0.099521   0.005315  18.723   <2e-16 ***
## age          0.012893   0.006006   2.147   0.0335 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3622 on 147 degrees of freedom
## Multiple R-squared:  0.7268, Adjusted R-squared:  0.7231 
## F-statistic: 195.5 on 2 and 147 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = voc_z ~ betaAgency + betaMachine, data = voc_REs_RL)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.13574 -0.23276 -0.02974  0.13115  1.68646 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.786072   0.056240 -13.977   <2e-16 ***
## betaAgency   0.101204   0.006202  16.319   <2e-16 ***
## betaMachine  0.001736   0.007136   0.243    0.808    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3678 on 147 degrees of freedom
## Multiple R-squared:  0.7184, Adjusted R-squared:  0.7145 
## F-statistic: 187.5 on 2 and 147 DF,  p-value: < 2.2e-16